Maximal Blocks in Morphic and Automatic Words

Let w be a morphic word over a finite alphabet Σ, and let ∆ be a subalphabet. We study the behavior of maximal blocks of ∆ letters in w, and prove the following: let (ik, jk) denote the starting and ending positions, respectively, of the k’th maximal ∆-block in w. Then lim supk→∞(jk/ik) is algebraic if w is morphic, and rational if w is automatic. As a results, we show that the same holds if (ik, jk) are the starting and ending positions of the k’th maximal zero block, and, more generally, of the k’th maximal x-block, where x is an arbitrary word. In this we answer a question posed by Yann Bugeaud.